0:00  
Hello, and thank you for listening to the mathematics teacher educator journal podcast. The mathematics teacher education journal is co sponsored by the Association of mathematics teacher educators, and the National Council of Teachers of Mathematics. My name is Eva Anheuser and I'm talking with Dr. Feldman, who is a clinical associate professor of mathematics education at the Wheelock College of Education and Human Development at Boston University. We will be discussing civs article encouraging teachers to make use of multiplicative structure published in the September 2018 issue of the mathematics teacher educator journal. We will begin by summarizing the main points of the article and discuss in more depth the lessons he shared in the article his successes and challenges and how these lessons relate to his other work. Ziv, thank you for joining us. Can you give us a brief summary of the article including the results?

0:58  
Sure. So the article and this article was written with Matt Roscoe from the University of Montana and myself. The article is really about a small study we did where we looked at 69 prospective elementary school teachers across two math content courses at our respective universities, we look to try to identify what is their understanding of prime factorization, specifically, their ability to use prime factorization as a tool for identifying divisibility, namely factors. So we had an intervention that we implemented in our courses, we saw the intervention consisted of three lessons and two homework assignments, we did a pre and post test. And what we found before the intervention was that by and large, the prospective teachers in our study kind of exhibited very clear kind of procedural attachments to their work around the visibility. In other words, they were not able to use prime factorization to determine the visibility, they reverted back to long division and trial and error type of methods. Whereas following the intervention, they were better able to use prime factorization not only to identify factors of numbers, but also to generate factors of numbers. So if I give you the prime factorization of a number, can you list out what its factors are? Can you list out not only the prime factors, but also any composite factor? So our study found that they were better able to not only do that, but also explain why, how they were using prime factorization, why that made sense?

2:35  
So why do we care about this?

2:37  
Well, I could tell you why I care about this. The reason why I think this is important is because of the focus on multiplicative structure. You know, we oftentimes in our content courses, we spend a lot of time talking about additive structures. When we talk about addition and subtraction, we, we decompose numbers additively in a variety of different ways in order to support computational strategies, to a lesser extent, we look at more look at of structure and trying to identify Okay, how do we decompose in order to support our perspective teachers understanding a variety of different strategies. This study, I think, provides more evidence that not only can pre service teachers do this work, but also that is important because it gives us an opportunity to make sense of a bunch of different ideas around multiplicative structure, namely commutative and associative properties, which they can leverage to make sense of additional topics that they might see down the road.

3:28  
Okay, who should read this article?

3:30  
I think this article might be relevant to a variety of people, primarily math teacher, educators, those who are charged with the preparation of prospective elementary school teachers, I think would find it most useful because the study provides curriculum materials that they could use in their own courses to support pre service teachers, understanding a multiplicative structure, but also anyone who is interested in better understanding how perspective elementary teachers develop knowledge of multiplicative structure. So those of us who do some research around how do prospective teachers learn mathematics? Or do they tend to understand where do they tend to find challenging? I think this study might be relevant for them as well.

4:09  
Okay, we talked a little bit about the why, but let's kind of talk about it again. What is the important problem or issue that your article addresses?

4:17  
I think, one, I don't know if it's an issue but but one thing that we see oftentimes in the literature on prospective teachers, mathematical knowledge, is example after example of what pre service teachers cannot do. From a mathematical perspective, we oftentimes read a lot of research around misconceptions that they have a challenges that they have in granted. This study was based on a lot of the research that that we read, in order to prepare for this study was just that, that pre service teachers have all these mathematical misconceptions. They have kind of these procedural attachments or leanings. They struggle to think kind of conceptually about big ideas. And so I think this study provides an example of kind of the opposite that that actually, prospective teachers can actually do pretty robust mathematics. And they can learn with understanding. And I think this article gives an example of just how that happens.

5:15  
So when I was reading your article, there's a strong focus on procedural fluency. And I was kind of curious how that connects to conceptual understanding, or whether that connects at all to conceptual understanding

5:32  
procedural fluency. You know, we in the article, and we go by definition that's provided by the National Research Council, and they're adding it up book. And so we kind of see the work in this study kind of firmly within kind of the procedural fluency umbrella, which is can you can you execute a procedure and not just rote Li but in a way where you understand what it is you're doing and why what you're doing makes sense. So I'm not exactly sure that we can say from this study that they've developed, you know, a deep conceptual understanding. But we certainly can say that based on the results that we found, the prospective teachers in the study are able to use prime factorization. And they're able to better understand the ways in which they're able to use prime factorization, why those ways make sense. And then one thing that we found, and maybe this kind of gets us a little bit closer to conceptual understanding is that through this study, we noticed that some folks in our sample were starting to reconceptualize a rethink their own definitions of what it means to be a factor of a number. So whereas previously, many of them would articulate a definition, that's kind of this classic factor of a number is a number that divides some other number or a number that if I multiply it by some whole number, I'll get kind of the number in question. They started through the intervention, they started thinking about a factor, kind of more generally as some multiplicative combination of prime numbers. And so if I have, for example, you know, the number 12, if I look at its prime factorization, you know, two times two times three, I can start to take multiplicative combinations of those prime factors to generate all the factors of the number 12. So for example, two, two times two, two times three, etc. And so once they started kind of rethinking their own definition of what it means to be a factor of a number, we started thinking, Oh, we, we might have something here, although we will need to do more research to figure out exactly how deep that new knowledge goes. But but we can definitely say that from a procedural standpoint, they not only were able to execute this procedure using prime factorization, a much better than they were before. In fact, before they weren't able to do it at all. But also they were able to articulate why using prime factorization actually allows you to find the factors of the number. And so we thought that that kind of procedural understanding was really highlighted in the study.

7:52  
I really liked your nod to the notion of the DNA of a number. Was that your idea? Or

7:59  
no, I can't take credit for that. That was some of my prospective teachers over the years came up with that kind of metaphor.

8:06  
I really like that that's helpful. So how does this article build on other work? You've done

8:11  
this article. And this study actually came out of an NSF grant that I'm a co pi on called the elementary mathematics project, which has been kind of a multi year project. The PI for that is Dr. Suzanne shapen, who's a colleague of mine at Boston University. And that project really was a curriculum development project where what we were doing is we were developing mathematical units curricular units for use in these prospective elementary teacher content courses. And one of the units that we developed was a unit on number theory. So we had, you know, each unit has about six to 10 lessons, this particular unit on number theory had about six or seven lessons that focused on kind of deconstructing multiplicative structure and, and trying to identify, okay, how do we determine the numbers divisibility? And by leveraging the structure of a number, how do we look at greatest common factors, least common multiples divisibility rules that we kind of all remember from our own kind of classroom experience, but maybe haven't thought exactly about how to justify those divisibility rules? That all came from that elementary mathematics project? This study, took some of those lessons, those lessons that we developed as kind of a starting point and revise them in order to kind of identify what are some of the key mathematical ideas that we want prospective teachers to grapple with just around determining divisibility? Just around this notion of multiplicative structure? While the elementary math project materials also address this topic, they were much more elaborate. So there were more lessons, each lesson was longer. We felt that we wanted to give math teacher educators kind of a A more digestible version of that, because we recognize that our syllabi and our content courses are already jam packed as it is. And so we wanted to give them just kind of the the maximum amount of lessons that we thought were important, so they could actually have the time to use them.

10:16  
So this actually leads really nicely into my next question, where I would like you to explain a little bit more about the innovations or the lessons and how they addressed this problem of practice.

10:29  
So we had three lessons and two homework assignments, I'll talk about the lessons, they're a little bit more interesting, essentially, what we try to do is we try to provide prospective teachers with kind of an image for what, you know, prime factorization is all about, and how can we determine the visibility, you know, not using kind of the traditional approaches of just kind of trial and error? Let's just do long division. And so the first lesson really focuses in on Okay, what do we mean by Prime factorization? We start by having prospective teachers draw factor trees, which many do remember from their prior kind of elementary and middle school experiences, you start to think about, okay, well, based on these factor trees, what can we say about the prime factorization of a number specifically about its its uniqueness, so we explore both in her spine factorization, but also a numbers factorization, which it has, could have many different factorizations. And so we get to this kind of really important foundational idea and number theory, which is the fundamental theorem of arithmetic. And that's really talking about how the prime factorization of a number is unique with the exception of kind of the order of the factors. And so we spend a lot of time in lesson one talking about prime factorization, talking about the uniqueness of the numbers, prime factorization, and that's where some of that those metaphors are on, you know, prime factorization as a numbers DNA, that's where that comes in. So they start to really get a sense of kind of prime numbers as the major building blocks of our number system. And then, you know, lesson two is really, so lesson one is kind of this foundational piece around let's develop this big idea around uniqueness of prime factorization. lesson two is where we kind of get into the nitty gritty around. Alright, so how do I use this notion of this prime factor representation to actually determine the visibility. And so you know, the previous, the first homework assignment of this intervention has use an array of one to 100, a 10 by 10 array. So essentially, they have first hundred numbers in this rectangular array. And what they're to do for homework is they're supposed to essentially find the prime factorization of the first hundred counting numbers and kind of draw them into the array. And then they bring that into lesson two. And then they use that in order to determine the factors of a variety of different numbers. Okay, and so they work in groups to do this. They're each group is assigned a number, they're using their their array, or their grid from the the first homework to kind of highlight using different colors, what are the factors of this number, we put everything up on the board, they create a table, and then from that they're able to start to generalize or draw some conclusions around. Okay, well, if they give you the prime factorization of a number, how can they use that to determine its factors? In other words, what seems to be common to all of the factors of a number when you look at them in their prime factored forms? And so folks start to realize, Oh, well, if I look at the prime factorization of a number, and then I look at the prime factorization of its factors, they all share prime numbers. And so they start to make that to realize that by looking at their array, their visual array, there's been some research in the past that talked about kind of having visual representations as cues for developing kind of a deeper understanding of kind of underlying structure. And so we tried to leverage that research and developing this array and specifically how we implement it into the lesson. So the lesson two kind of goes into that lesson three, then culminates with a kind of an activity where they tried to generalize their experience from lesson two. And they specifically talked about, you know, how can we use prime factorization to determine how many factors the number has, so not necessarily what the factors are? They've already done that. Now, how many are then so they generate some data, they start out by listing out several examples of numbers that have two factors, three factors, four factors that go all the way up to six. And then they examine all those numbers in prime factored form. And from there, they attempt to generalize this idea that Oh, if I, if I look at the numbers prime factor form, I can very easily determine how many factors it has by looking at the exponents. You know, add one to each exponent, then multiply those sums together. And I'll figure out how many factors I have. And so that's kind of an exercise in using what we've done around prime factorization and generalizing that collecting data, making conjectures refining conjectures, and then generalizing to all cases.

15:01  
I really liked how you started off the first lesson with just having them take the same number and create different prime factorizations. I mean, different different ordered prime factorizations. Okay, so thanks for this awesome description of the lessons Now how did you study whether they worked,

15:22  
so what we did is we created a an assessment, we created a pretest, and then a corresponding post test. But we administered the pretest, immediately before the intervention. So about a week before, we started these three lessons and two homework assignments. And then we gave the post test about up to a week after. And based on those, and then we analyze the results of those, essentially, what we did is both Matt and I, as co authors, we've looked at all the data we created are a scoring rubric, we each in order to develop inter rater reliability, we kind of took a subsample of the of the data we scored a common sub sample independently came back together we had, you know, I think we reached about something like 80 to 83% agreement, and then any discrepancies, we kind of would spend a lot of time kind of talking to each other about Okay, well, how did you interpret the rubric? How did you interpret the rubric, and then eventually got to 100% agreement. And once we got to 100% agreement, then we split off and essentially scored and coded the rest of the sample

16:32  
and summarize briefly the findings of that analysis.

16:37  
Yeah, sure. So essentially, there were, you know, in the article, there's kind of this overarching question around, you know, what's the impact of the intervention on pre service teachers kind of procedural knowledge as it relates to factors in prime factorization. So the overall data kind of showed that, overwhelmingly, their ability to use prime factorization to determine divisibility improved significantly, it was overwhelming. But the really interesting part isn't really a we kind of assumed that that would happen. Typically, you have an intervention in your class, you collect data before and after you're going to see positive results. It's the nature of the change that we thought was compelling. And so kind of one of the questions we were one of the specific questions we were asking, is, can pre service teachers in the sample, can they use a numbers prime factorization to identify a variety of different types of numbers? So can they use them to successfully identify prime factors, prime non factors, composite factors, composite non factors, and we found that they were they improved in their ability to do all of those. But one thing was really interesting is that in this, the results that we found really mirrored what prior research has also shown, when I talked about prior research in this space, I'm talking about primarily the work of Rena Vasquez in the 90s, where she looked at prospective elementary teachers knowledge of prime factorization. And she found that, you know, they had a much better and much stronger procedural knowledge of using prime factorization to identify prime factors, but not really as easy of a time identifying composite factors. And similarly, it was much easier for pre service teachers to identify factors than non factors. And what our results showed is that this actually held true on the pretest. Is that exactly the same thing happened. However, on the post test, our entire sample improved significantly. And so it got to the point where their ability based on the data to identify both prime factors and prime non non factors is almost indistinguishable, which means something was going on that impacted their ability to identify both the visibility and indivisibility and prior research kind of talks about indivisibility is is kind of a challenge for prospective teachers. And likewise, also composite factors and composite non factors, they all improve the noun as well. So that was really great. The other question that we're really wanting to know is, did the intervention really improve their ability to use prime factorization to generate their own list of factors? So instead of just identifying, you know, if I give you a number, you tell me if it's a factor or not, in this situation, now it's I give you a prime factorization, and you have to generate the list of factors yourself. And we saw that improved significantly there as well. And the way we did this is we asked them two types of questions. The first type of question we asked them, you know, finding the factors of a number and we gave them the number and kind of decimal form. So I think the number we use, there was, I think, 225 or 300. And we wrote it in decimal form. And then the second part, we asked him the same question, but we gave them a different number in factored form. And what we found before the intervention is that they were much more successful at finding the factors when the number was represented in decimal form. Still wasn't great but It was almost Well, I would say almost twice as successful, but it was much more successful. And then on the post test, they improved significantly, regardless of the numerical representation, but to the point where the success rate was almost indistinguishable was around 70% for both of them. So still work to be done. But something happened during this intervention, where their ability to use either decimal form or prime factored form was essentially the same. And that's not what prior research tells us, it tells us that, you know, we're much more comfortable with looking at a whole number forms, we can use long division very, very successfully. In this case, prime factor form seemed to be just as I shouldn't say, just as easy. But the data tells us that they had as much success at it as with decimal form, so that was kind of the second finding. And then the third finding that I found most interesting, as we asked them, too, we gave them some properties of numbers, specifically, we told them, hey, find me a number that has these factors. And there were so essentially, they were reversing the procedures that they were doing before, instead of me giving them the number and asked them to find the factors, we give them the factors that Hey, tell me what number has these factors. And that's much more challenging, because they're actually essentially reversing the procedure that they've been accustomed to. And we found that that was before the intervention extremely challenging for our prospective teachers. In fact,

21:27  
it was not a single participant in the entire sample that was able to both correctly find the answer, and also explain and justify their thinking. And in fact, only I think about 15, or 16% of the sample could find me the correct answer. On post test, we found that this changed dramatically. And I think when happening is I think, roughly three quarters of the sample on the post test found the correct answer, they're able to reverse this procedure. And then about half of the sample, we're able to also provide kind of what we called complete or nearly complete reasoning. And so what was going on is, in this finding, we saw they were able to really start coordinating prime factors together. So when you look at a numbers, prime factorization, you can identify composite factors by taking multiple primes, and essentially combining them together. And on this problem, we saw that that was, that had to happen, they had to do that. And so in order to be successful, we saw that that was happening a lot more readily than before the intervention. So overall, the improvement was significant across the board, but it's kind of the nature of the improvement that we really found compelling.

22:39  
So sounds like that, that well, connectedness really came out into post test,

22:45  
it seems like it, you know, and there's, I think there's still more work to be done to really kind of unpack, okay, how, what was the nature of their thinking around this third problem, specifically, but in all the written work that we analyzed, we found that about half of the sample was able to really articulate not only how they solve this problem, how they're able to connect and coordinate these prime factors together, but also why that made sense. So the justification piece made sense, you know, they, they would say things like, I was able to find, you know the answer by combining these prime factors, because I know that the factors of a number are multiplicative combinations of its prime factors. And so they're able to relate it back to this kind of bigger idea on what it means to be a factor. And so we found that to be really encouraging, and really exciting. Because, you know, as we know, a lot of the research around prospective teacher knowledge highlights their misconceptions and the challenges they face. And we found, oh, this is a situation where now we have this intervention that based on the data is supporting them to kind of develop this richer knowledge.

23:54  
So that kind of leads into the next question, what new contribution to our field of math teacher educators does your article make? And I think you just summarized some of that. But I'll yeah,

24:04  
I can kind of restate it in a slightly different way. I think they're kind of two primary things. One is we're providing curriculum materials, then math teacher educators can use in their courses, by no means are the three lessons and two homework assignments comprehensive. But I think it's a good start. And I think the data tells us it's a good start. But the other thing is, I think the contribution is around, you know, teacher knowledge is specifically perspective, teacher knowledge around how do we develop an understanding for how future teachers make sense of the mathematics and you know, as well as anyone that there's a lot of research out there around that. And I think we're contributing to the field around multiplicative structure, because there's not a lot of research out there that I know of, aside from the work that Venus ask us is done. That really unpacks what to pre service teachers know about prime numbers, about prime factorization about the visibility and about how those three topics are connected. to one another. So I think this study gets us started on that road and, and then tells us that, hey, pre service teachers can do really, really good mathematical work, they, you know, they can shed misconceptions if they're provided opportunities to do so. And we think based on this study, that having them look at particular representations like kind factorization, and this instance, kind of almost forcing them to, you know, not necessarily abandon whole number representations, but to kind of put them aside for a minute and think about a different way of looking at number two be really beneficial, and it kind of forces them into the state of uncertainty, we think that can be productive.

25:44  
If you imagine that other people use the lessons that you attached to your article, which I definitely plan on doing, how do you see other people using the lessons? And what would you like to hear back from people who are using your lessons,

25:58  
I'm under no illusion that different people will, who use these lessons will use them differently. And I think that's perfectly fine. The lessons aren't meant to be prescriptive, I would like to hear that folks are using the one to 100 grid, we found and we haven't really collected data yet on the use of the grid, per se. But, you know, just anecdotally, we felt that when you're in the classroom working with prospective teachers, and they're using this grid, and they're shading in the different squares that represent the factors of whatever number they're exploring, that they can refer back to that grid that helped them kind of make sense of, and help them identify, okay, what do the prime factorizations of the factors of a number and the number itself have in common, we thought that was really useful. So we hope that folks would consider using the array or the grid, but really what I would love to hear feedback from from other math teacher educators who choose to use this, you know, we're always looking to revise and modify tasks to make them more effective and more relevant for our students. And so we'd love to hear feedback from people, you know, I know different people will, will use these in different ways. And so I guarantee you, they're different approaches that we haven't thought about, we'd love to hear from folks.

27:14  
That's fantastic. So now that people have heard your podcast and read your article, and are very excited about dork, what's your suggestion?

27:22  
Honestly, I think that one suggestion is not as much a suggestion as it is a hope is that folks who are kind of charged with the preparation or prospective teachers will look in materials like these or design materials like these, in order to provide their prospective teachers with more opportunities to kind of grapple with mathematical ideas, and to give them time to actually kind of think through these challenging topics. I know that, you know, we have, you know, I think all of us, you know, we create our syllabi for these content courses, and we we pack them with all sorts of content that we know our prospective teachers need to know. And sometimes what might get lost in the shuffle is kind of giving our students the time to kind of work together to share ideas, to test their conjectures, revise them, that all takes time. And then to then share their ideas out as a class and doing a running a classroom discussion with a room of 20 to 30 students is challenging work, it takes a lot of time. But I would hope that we as a, as a community of math teacher educators know many of us do this already. I think that's critical. If we want to engage our prospective teachers in these kinds of activities, we need to make the time to do that. And so sometimes that means maybe eliminating some topics from from your syllabus, which I know is was not supposed to say that out loud. But that's something that I would hope people would would consider doing. And I know that many people already do.

28:52  
I really enjoyed the examples you gave in the paper. And I find those helpful when I imagine teaching these lessons. So I plan on doing it and I'll plan on letting you know whether they worked or not.

29:05  
Fantastic. I look forward to your feedback. Thank you.

29:08  
All right. Well, thank you for joining us and good bye yeah.